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Mar 15, 2016
03/16

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May 1, 2017
05/17

May 1, 2017
by
Kolosov Petro

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Abstract . Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown....

Topics: Power function, Monomial, Polynomial, Power series (mathematics), Exponential function, Power...

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Apr 30, 2017
04/17

Apr 30, 2017
by
Kolosov Petro

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Abstract . The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials. MSC 2010 : 46G05, 30G25, 39-XX a rXiv :1608.00801 DOI : 10.6084/m9.figshare.4955384 Keywords : Power function, Monomial, Polynomial, Power series (mathematics), Exponential function, Power (mathematics), Exponentiation, Mathematical Series, Cube...

Topics: Power function, Monomial, Polynomial, Power series (mathematics), Exponential function, Power...

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22

Apr 30, 2017
04/17

Apr 30, 2017
by
Kolosov Petro

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Abstract . This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, based on induction from particular to general case, except above described theorems. MSC 2010 : 40C15, 32A05 arXiv...

Topics: Mathematics, power function, monomial, polynomial, power series, series, finite difference, divided...

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37

Jul 8, 2016
07/16

Jul 8, 2016
by
Kolosov Petro

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Abstract . This paper presents the way to make expansion for the next form function: y =x^n , ∀(x,n) ∈ N to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described theorems. MSC 2010 : 40C15, 32A05 arXiv : 1603.02468 DOI : 10.6084/m9.figshare.3475034 Keywords : Power function,...

Topics: Power function, Monomial, Polynomial, Power series (mathematics), Exponential function, Power...

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32

Apr 7, 2016
04/16

Apr 7, 2016
by
Kolosov Petro

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Abstract . This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described theorems. MSC 2010 : 40C15, 32A05 arXiv : 1603.02468 DOI : 10.6084/m9.figshare.3475034 Keywords :...

Topics: Mathematics, power function, monomial, polynomial, power series, series, finite difference, divided...

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Mar 16, 2016
03/16

Mar 16, 2016
by
Kolosov Petro

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This paper presents the way to make expansion for the next form function: y = x^n, ∀(x, n) ∈ N to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described theorems. MSC 2010 : 40C15, 32A05 arXiv :1603.02468 Keywords : power, power function, monomial, polynomial, power series, third...

Topics: number to power, cube number, series representation, series expansion, binomial theorem, power...